任意深度指数剪切流不稳定性产生的自由水面波浪

M. Abid, C. Kharif
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引用次数: 0

摘要

利用韦伯数和弗劳德数重新研究和阐述了水中指数流在重力和毛细作用下对无穷小扰动的稳定性。本文提出了一些关于重力-毛细管波产生的新结果,补充了莫兰等人之前的研究成果["Waves generated by shear layer instabilities," Proc.Math.433, 441-450 (1991)] 以及 Young 和 Wolfe ["Generation of surface waves by shear-flow instability," J. Fluid Mech.739, 276-307 (2014)]的有限深度。为了考虑更大尺度的扰动,我们特别关注了指数流在重力作用下的稳定性。更确切地说,本研究揭示了不同环境条件下指数剪切流稳定性的重要见解。值得注意的是,我们发现无量纲增长率随着弗劳德数的增加而增加,从而加深了对剪切层厚度和表面速度之间相互作用的理解。此外,我们的分析还阐明了最不稳定模式的尺寸波长,强调了其与特征剪切层厚度的相关性。此外,在重力-毛细管不稳定性领域,我们根据韦伯数建立了指数流稳定性的充分条件。我们的发现得到了有限深度稳定性图的支持,该图显示了稳定域的大小与剪切层特征厚度的相关性。此外,我们还探索了指数剪切流中液体薄膜的稳定性,进一步丰富了我们对此类系统中复杂动力学的理解。
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Free surface water waves generated by instability of an exponential shear flow in arbitrary depth
The stability of an exponential current in water to infinitesimal perturbations in the presence of gravity and capillarity is revisited and reformulated using the Weber and Froude numbers. Some new results on the generation of gravity-capillary waves are presented, which supplement the previous works of Morland et al. [“Waves generated by shear layer instabilities,” Proc. Math. Phys. Sci. 433, 441–450 (1991)] and Young and Wolfe [“Generation of surface waves by shear-flow instability,” J. Fluid Mech. 739, 276–307 (2014)] on finite depth. To consider perturbations at much larger scales, special attention is given to the stability of exponential currents only in the presence of gravity. More precisely, the present investigation reveals significant insights into the stability of exponential shear currents under different environmental conditions. Notably, we have identified that the dimensionless growth rate increases with the Froude number, providing a deeper understanding of the interplay between shear layer thickness and surface velocity. Furthermore, our analysis elucidates the dimensional wavelength of the most unstable mode, emphasizing its relevance to the characteristic shear layer thickness. Additionally, within the realm of gravity-capillary instabilities, we have established a sufficient condition for the stability of exponential currents based on the Weber number. Our findings are supported by stability diagrams at finite depth, showing how the size of stable domains correlates with the characteristic thickness of the shear layer. Moreover, we have explored the stability of a thin film of liquid in an exponential shearing flow, further enriching our understanding of the complex dynamics involved in such systems.
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